Important Questions for CBSE Class 12 Maths Chapter 1 – Relations and Functions

CBSE Class 12 Maths Chapter-1 Important Questions – Free PDF Download

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CBSE Class 12 Mathematics Important Questions Chapter 1 Relations and Functions


1 Mark Questions

1. A Relation R:AàA is said to be Reflexive if ——— for every a  A where A is non
empty set.
Ans: (a, a) R


2. A Relation R:AàA is said to be Symmetric if ———-  a,b, A
Ans: (a, b) R, (b, a) R


3. A Relation R:AàA is said to be Transitive if ————-  a,b,c  A
Ans: (a, b)R, and (b, c)(a, c) R.


4. Define universal relation? Give example.
Ans: A Relation R in a set A called universal relation if each element of A is related to every element of A. Ex. Let = {2,3,4}
R = (AA) = {(2,2),(2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4) }


5. What is trivial relation?
Ans: Both the empty relation and the universal relation are some time called trivial relation.


6. Prove that the function f: R à R, given by f(x) = 2x, is one – one.
Ans: f is one – one as f(x1) = f (x1)
2x= 2x2
x1 = x2
Prove.


7. State whether the function is one – one, onto or bijective f: R à R defined by f(x) = 1+ x2
Ans: Let x1, x2 x
If f(x1) = f(x2)


Hence not one – one


8. Let S = {1, 2, 3}
Determine whether the function f: S à S defined as below have inverse.
f = {(1, 2), (2, 1), (3, 1)}
Ans: f(2) = 1 f(3) = 1,
f is not one – one, So that that f is not invertible.


9. Find gof f(x) = |x|, g(x) = |5x + 1|
Ans: gof (x) = g [f(x)]
= g [(x)]


10. Let f, g and h be function from R to R show that (f + g) oh = foh + goh
Ans: L.H.S = (f + g) oh
= {(f + g) oh} (x)
= (f + g) h (x)
= f [h (x)] + g [h (x)]
= foh + goh


9. If a * b = a + 3b2, then find 2 * 4
Ans: 2 * 4 = 2 + 3 (4)2
= 2 + 3 16
= 2 + 48
= 50


11. Show that function f: N à N, given by f(x) = 2x, is one – one.
Ans: the function f is one – one, for
f(x1) = f(x2)
2x1 = 2x2
x1 = x2


12. State whether the function is one – one, onto or bijective f: R à R defined by f(x) = 3 – 4x
Ans: is x1, x2 R
f(x1) = f(x2)
3 – 4x1 = 3 – 4x2
x1 = x2
Hence one – one
Y = 3 – 4x

= y
Hence onto also.


13. Let S = {1, 2, 3}
Determine whether the function f: S à S defined as below have inverse.
f = {(1, 1), (2, 2), (3, 3)}
Ans: f is one – one and onto, so that f is invertible with inverse f-1 = {(1, 1) (2, 2) (3, 3)}
14. Find got f(x) = |x|, g(x) = |5x -2|
Ans: fog (x) = f(g x)
= f{|5x – 2|)
= |5x – 2|


15. Consider f: {1, 2, 3} à {a, b, c} given by f(1) = a, f(2) = b and f(3) = c find f-1 and show
that (f-1)-1 = f
Ans: f = {(1, a) (2, b) (3, c)}
f-1 = { (a, 1) (b, 2) (c, 3)}
(f -1-1 = {(1, a) (2, b) (3, c)}
Hence (f-1)-1 = f.


16. If f(x) = x + 7 and g(x) = x – 7,  x R find (fog) (7)
Ans: (fog) (x) = f[g(x)]
= f(x – 7)
= x – 7 + 7
= x
(fog) (7) = (7)


17. What is a bijective function?
Ans: A function f: X à Y is said to be one – one and onto (bijective), if f is both one – one and onto.


18 Let f: R à R be define as f(x) = x4 check whether the given function is one – one onto,
or other.
Ans: Let x1, x2 R
If f(x1) = f(x2)

Not one – one

Not onto.


19 Let S = {1, 2, 3}
Determine whether the function f: S à S defined as below have inverse.
f = {(1, 3) (3, 2) (2, 1)}
Ans: f is one – one and onto, Ao that f is invertible with f-1 = {(3,1) (2, 3) (1, 2)}
20 Find gof where f(x) = 8x3, g(x) = x1/3
Ans: gof (x) = g[f(x)]
= g (8x3)

= 2x


21. Let f, g and h be function from R + R. Show that (f.g) oh = (foh). (goh)
Ans: (f. g) oh
(f. g) h (x)
f[h(x)]. g[h(x)]
foh. goh


21. Let * be a binary operation defined by a * b = 2a + b – 3. find 3 * 4
Ans: 3 * 4 = 2 (3) + 4-3 = 7


22. show that a one – one function f: {1, 2, 3} à {1, 2, 3} must be onto.
Ans: Since f is one – one three element of {1, 2, 3} must be taken to 3 different element of the co – domain {1, 2, 3} under f. hence f has to be onto.


23. f: R à R be defined as f(x) = 3x check whether the function is one – one onto or other
Ans: Let 



24. Let S = {1, 2, 3}
Determine whether the function f: S à
 S defined as below have inverse.
f = { (1, 2) (2, 1) (3, 1) }

Ans:f(2) = 1, f(3) =1
f is not one – one so that f is not invertible
Hence no inverse


25. Find fog f(x) = 8x3, g(x) = x1/3
Ans: fog (x) = f(gx)

= 8x


26. If f: R à R be given by f(x) =  , find fof (x)
Ans: 

= x


27. If f(x) is an invertible function, find the inverse of f(x) = 
Ans: Let f(x) = y



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4 Marks Questions

1. Let T be the set of all triangles in a plane with R a relation in T given by
R = {(T1, T2): T1 is congruent to T2}.
Show that R is an equivalence relation.
Ans. R is reflexive, since every  is congruent to itself.
(T1T2)R similarly  (T2T1) R
since T1 T2
(T1T2) R, and (T2,T3) R
(T1T3)R Since three triangles are
congruent to each other.


2. Show that the relation R in the set Z of integers given byR={(a, b) : 2 divides a-b}. is equivalence relation.
Ans. R is reflexive , as 2 divide a-a = 0
((a,b)R ,(a-b) is divide by 2
 (b-a) is divide by 2 Hence (b,a) R hence symmetric.
Let a,b,c   Z
If  (a,b) R
And (b,c)  R
Then  a-b and b-c is divided by 2
 a-b +b-c is even
 (a-c is even
 (a,c) R
Hence it is transitive.


3. Let L be the set of all lines in    plane and R be the relation in L define if R = {(l1, L2 ): L1 is   to L2 } . Show that R is symmetric but neither reflexive nor transitive.
Ans. R is not reflexive , as a line L1 cannot be  to itself  i.e  (L1,L1 ) R
 L1 L2
L2 L1
 (L2,L1)R
L1  L2 and L2 L3

 
Then L1 can never be   to L3 in fact L1 || L3
i.e (L1,L2) R, (L2,L3)  R.
But (L1, L3) R


4. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a+1} is reflexive, symmetric or transitive.
Ans. R = {(a,b): b= a+1}
Symmetric or transitive
R = {(1,2) (2,3) (3,4) (4,5) (5,6) }
R is not reflective , because (1,1) R
R is not symmetric because (1,2) R but (2,1) R
(1,2) R and (2,3) R
But (1,3) R Hence it is not transitive


5. Let L be the set of all lines in xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.
Find the set of all lines related to the line y=2x+4.
Ans. L1||L1    i.e (L1, L1)  R  Hence reflexive
L1||L2 then L2 ||L1   i.e (L1L2) R
 (L2,L) R Hence symmetric
We know the
L1||L2  and L2||L3
Then L1|| L3
Hence Transitive .  y = 2x+K
When K is real number.


6. Show that the relation in the set R of real no. defined R = {(a, b) : a  b3 }, is neither reflexive nor symmetric nor transitive.
Ans. (i) (a, a)   Which is false R is not reflexive.
(ii)  Which is false R is not symmetric.
(iii)  Which is false


7. Let A = N N and * be the binary operation on A define by (a, b) * (c, d) = (a + c, b + d)Show that * is commutative and associative.
Ans. (i) (a, b) * (c, d) = (a + c, b + d)
= (c + a, d + b)
= (c, d) * (a, b)
Hence commutative
(ii) (a, b) * (c, d) * (e, f)
= (a + c, b + d) * (e, f)
= (a + c + e, b + d + f)
= (a, b) * (c + e, d + f)
= (a, b) * (c, d) * (e, f)
Hence associative.


8. Show that if f:   is defining by f(x) =  and g:  is define by
g(x) =  then fog = IA and gof = IB  when  ; IA (x) = x, for all x A, IB(x) = x, for all x B are called identify function on set A and B respectively.
Ans. gof (x) = 


Which implies that gof = IB
And Fog = IA


9. Let f: N  à N be defined by f(x) = 
Examine whether the function f is onto, one – one or bijective
Ans. 

f is not one – one
1 has two pre images 1 and 2
Hence f is onto
f is not one – one but onto.


10. Show that the relation R in the set of all books in a library of a collage given by R ={(x, y) : x and y  have same no of pages}, is an equivalence relation.
Ans. (i)  (x, x)  R, as x and x have the same no of pages for all x R is reflexive.
(ii)  (x, y) R
x and y have the same no. of pages
y and x have the same no. of pages
(y, x) R
(x, y) = (y, x) R is symmetric.
(iii)  if (x, y)  R, (y, y)  R
(x, z) R
 R is transitive.


11. Let * be a binary operation. Given by a * b = a – b + abIs * :
(a) Commutative
(B) Associative
Ans. (i)  a * b = a – b + ab
b * a = b – a + ab
a * b  b * a
(ii)  a * (b * c) = a * (b – c + bc)
= a – (b – c + bc) + a. (b – c + bc)
= a – b + c – bc + ab – ac + abc
(a * b) * c = (a – b + ab) * c
= [ (a – b + ab) – c ] + ( a – b + ab)
= a- b + ab – c + ac – bc + abc
a * (b * c)   (a * b) * c.


12. Let f: R à R be f (x) = 2x + 1 and g: R à R be  g(x) = x2 – 2 find (i) gof (ii) fog
Ans. (i)  gof (x) = g[f(x)]
= g (2x + 1)
= (2x + 1)2 – 2
(ii)  fog (x) = f (fx)
= f (2x + 1)
= 2(2x + 1) + 1
= 4x + 2 + 1 = 4x + 3


13. Let A = R – {3} and B = R- {1}. Consider the function of f: A à B defined by
f(x) =   is f one – one and onto.
Ans. Let x1 x2  A
Such that f(x1) = f(x2)

f is one – one


Hence onto


14. Show that the relation R defined in the set A of all triangles asR = {  is similar to T2 }, is an equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5. T2 with
sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
Ans. (i)  Each triangle is similar to at well and thus (T1, T1)  R
  R is reflexive.
(ii)  (T1, T2)  R
 T1 is similar to T2
T2 is similar to T1
(T2, T1)  R
R is symmetric
(iii)  T1 is similar to T2 and T2 is similar to T3
T1 is similar to T3
(T1, T3)  R
R is transitive.
Hence R is equivalence
(II) part  T1 = 3, 4, 5
T2 = 5, 12, 13
T3 = 6, 8, 10
T1 is relative to T3.


15. Determine which of the following operation on the set N are associative and which are commutative.
(a) a * b = 1 for all a, b  N
(B) a * b =   for all a, b,  N
Ans. (a)  a * b = 1
b * a = 1
for all a, b  N also
(a * b) * c = 1 * c = 1
a * (b * c) = a * (1) = 1 for all, a, b, c R N
Hence R is both associative and commutative
(b)  a * b = ,  b * a = 
Hence commutative.
(a * b) * c = 
=


* is not associative.


17. Let A and B be two sets. Show that f: A  B à B  A such that f(a, b) = (b, a) is a bijective function.
Ans. Let (a1 b1) and (a2, b2)  A  B
(i)  f(a1 b1) = f(a2, b2)
b1 = b2 and a1 = a2
(a1 b1) = (a2, b2)
Then f(a1 b1) = f(a2, b2)
(a1 b1) = (a2, b2) for all
(a1 b1) = (a2, b2)  A  B
(ii)  f is injective,
Let (b, a) be an arbitrary
Element of B  A. then b  B and a  A
(a, b) )  (A  B)
Thus for all (b, a)  B  A their exists (a, b) )  (A  B)
Hence that
f(a, b) = (b, a)
So f: A  B à B  A
Is an onto function.
Hence bijective


18. Show that the relation R defined by (a, b) R (c, d)  a + b = b + c on the set N  N is an equivalence relation.
Ans. (a, b) R (c, d)  a + b = b + c where a, b, c, d  N
(a, b ) R (a, b)  a + b = b + a (a, b)  N  N
R is reflexive
(a, b) R (c, d) a + b
= b + c (a, b ) (c, d)  N  N
d + a = c + b
c + b = d + a
 (c, d) R (a, b) (a, b), (c, d)  N  N
Hence reflexive.
(a, b) R (c, d)  a + d = b + c   (1) (a, b), (c, d)  N  N
(c, d) R (e, f)  c + f = d + e   (2) (c, d), (e, f)  N  N
Adding (1) and (2)
(a + b) + [(+f)] = (b + c) + (d + e)
a + f = b + e
(a, b) R (e, f)
Hence transitive
So equivalence


19. Let * be the binary operation on H given by a * b = L. C. M of a and b. find
(a) 20 * 16
(b) Is  * commutative
(c) Is * associative
(d) Find the identity of * in N.
Ans. (i)  20 * 16 = L. C.M of 20 and 16
= 80    
 
(ii)  a * b = L.C.M of a and b
= L.C.M of b and a
= b * a
(iii)  a * (b * c) = a * (L.C.M of b and c)
= L.C.M of (a and L.C.M of b and c)
= L.C.M of a, b and c
Similarity
(a * b) * c = L. C.M of a, b, and c
(iv)  a * 1 = L.C.M  of a and 1= a
=1


20. If the function f: R à  R is given by f(x) =   and g: R à R is given by g(x) = 2x – 3, Find 
(i) fog  
(ii) gof. Is f-1 = g
(iii)  fog = gof = x
Ans. (i)  fog (x) = f [g(x)]
= f (2x – 3)

= x
(ii)  gof (x) = g [f(x)]

= x
(iii)  fog = gof = x
Yes,


21. Let L be the set of all lines in Xy plane and R be the relation in L define as R = {(L1, L2): L1 || L2} Show then R is on equivalence relation.
Find the set of all lines related to the line y=2x+4.
Ans. L1||L1    i.e (L1, L1)  R  Hence reflexive
L1||L2 then L2 ||L1   i.e (L1L2) R
 (L2, L) R Hence symmetric
We know the
L1||L2 and L2||L3
Then L1|| L3
Hence Transitive.  y = 2x+K
When K is real no.

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